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A particle of mass ' $\mathrm{m}$ ' is kept at rest a height $3 \mathrm{R}$ from the surface of earth, where ' $R$ ' is radius of earth and ' $M$ ' is the mass of earth. The minimum speed with which it should be projected upward, so that it does not return back is
( $\mathrm{g}=$ acceleration due to gravity on the earth's surface)
Options:
( $\mathrm{g}=$ acceleration due to gravity on the earth's surface)
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1519 Upvotes
Verified Answer
The correct answer is:
$\left[\frac{\mathrm{GM}}{2 \mathrm{R}}\right]^{1 / 2}$
$$
\mathrm{K}_{\mathrm{i}}+\mathrm{U}_{\mathrm{i}}=\mathrm{K}_{\mathrm{f}}+\mathrm{U}_{\mathrm{f}}
$$
$$
\frac{1}{2} \mathrm{mv}^2-\frac{\mathrm{GMm}}{4 \mathrm{R}}=0
$$
$$
\therefore \mathrm{v}=\left[\frac{\mathrm{GM}}{2 \mathrm{R}}\right]^{1 / 2}
$$
\mathrm{K}_{\mathrm{i}}+\mathrm{U}_{\mathrm{i}}=\mathrm{K}_{\mathrm{f}}+\mathrm{U}_{\mathrm{f}}
$$
$$
\frac{1}{2} \mathrm{mv}^2-\frac{\mathrm{GMm}}{4 \mathrm{R}}=0
$$
$$
\therefore \mathrm{v}=\left[\frac{\mathrm{GM}}{2 \mathrm{R}}\right]^{1 / 2}
$$
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